We define the unit circle as the collection of all vectors with length 1 centered at some point. (The one below specifically defines the unit circle centered at the origin)
$$\mathscr{C} = \{ x\in\mathbb{R}^2 \colon \|\|x\|\| = 1 \}.$$
In this case, the norm used is the Euclidean norm $p = 2$. But we may use the $\ell_p$ metric for $p\in[1,\infty)$ where
$$\|\|x\|\|_p = \left(\|x_1\|^p + \|x_2\|^p\right)^{1/p}.$$
We use these metrics to generalize the unit circle as the $\ell_p$ unit circles:
$$\mathscr{C}_p = \{ x\in\mathbb{R}^2 \colon \|\|x\|\|_p = 1 \} .$$
I have depicted some unit circles from $p\in[1,\infty)$.
$\ell_p$ Unit Circles" />My question is, what would the group action be on these circles? Of course, the Euclidean circle is part of $O(2)$, but that falters when $p$ deviates from 2 (is this trivial since orthogonal groups preserve the Euclidean norm only?) We see that there are lines of symmetry about $y = 0$, $y=x$, $y=-x$, $x=0$. I'm new to group theory, but would we say that the dihedral group $Dih_4$ acts on $\mathscr{C}_p$?
Richter provides generalized trigonometric functions for $\ell_p$ metric spaces. Specifically,
$$\cos_p(\theta) = \frac{\cos(\theta)}{N_p(\theta)}$$
$$\sin_p(\theta) = \frac{\sin(\theta)}{N_p(\theta)}$$
where, $N_p(\theta) = \left(\|\cos(\theta)\|^p + \|\sin(\theta)\|^p\right)^{1/p}.$ These generalized trigonometric functions can also parametrize the curve of $\mathscr{C}_p$ as $r_p(\theta) = (\cos_p(\theta), \sin_p(\theta))$. Certainly if $N_p(\theta) = 1$, then we have our Euclidean trigonometric functions. I created a contour plot ($p$ vs. $\theta$) for what angles $N_p(\theta) = 1$ is valid below.
$N_p(\theta) = 1$ for $p$ vs $\theta$ " />
The plot roughly shows that when $p\neq2$ this occurs when $\theta \in \{0, \pi/2, \pi, 3\pi/2\}$ which conforms with the lines of symmetry above. How do I concretely prove that these are the only lines of symmetry and demonstrate/describe the valid group action on $\mathscr{C}_p$?
EDIT: I know that my contour plot is also trying to demonstrate that the determinant of this matrix is equal to 1,
$R=\begin{bmatrix}\cos_p(\theta)&-\sin_p(\theta)\\\cos_p(\theta)&\sin_p(\theta)\end{bmatrix}$
or that $R^T = R^{-1}$ iff $\det(R) = 1$.